Substituting into the Second Equation

Oct 13, 2025 by miodeya.com 34 views

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.

The System of Equations

The given system of equations is:

${ \begin{aligned} -3x + 6y &= 9 \\ 5x + 7y &= -49 \end{aligned} }$

Method of Substitution

To solve this system of equations, we can use the method of substitution. We will solve one equation for one variable and then substitute that expression into the other equation.

Let's solve the first equation for x:

${ -3x + 6y = 9 }$

Subtracting 6y from both sides:

${ -3x = 9 - 6y }$

Dividing both sides by -3:

${ x = \frac{9 - 6y}{-3} }$

Simplifying:

${ x = -3 + 2y }$

Substituting into the Second Equation

Now, we will substitute the expression for x into the second equation:

${ 5x + 7y = -49 }$

Substituting x = -3 + 2y:

${ 5(-3 + 2y) + 7y = -49 }$

Expanding:

${ -15 + 10y + 7y = -49 }$

Combining like terms:

${ -15 + 17y = -49 }$

Adding 15 to both sides:

${ 17y = -34 }$

Dividing both sides by 17:

${ y = -2 }$

Finding the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations. Let's use the first equation:

${ -3x + 6y = 9 }$

Substituting y = -2:

${ -3x + 6(-2) = 9 }$

Expanding:

${ -3x - 12 = 9 }$

Adding 12 to both sides:

${ -3x = 21 }$

Dividing both sides by -3:

${ x = -7 }$

Conclusion

Therefore, the solution to the system of equations is x = -7 and y = -2.

Checking the Solution

To verify that this solution is correct, we can substitute x = -7 and y = -2 into both original equations:

${ -3x + 6y = -3(-7) + 6(-2) = 21 - 12 = 9 }$

${ 5x + 7y = 5(-7) + 7(-2) = -35 - 14 = -49 }$

Both equations are satisfied, so we can conclude that the solution is correct.

Answer

The correct answer is:

B. $(-7, -2)$

Discussion

This problem involves solving a system of two linear equations with two variables. We used the method of substitution to find the solution. The key steps involved solving one equation for one variable, substituting that expression into the other equation, and then solving for the other variable. We also verified the solution by substituting the values back into both original equations. This problem requires a strong understanding of algebraic concepts, including solving linear equations and systems of equations.

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. In our previous article, we discussed the method of substitution and elimination to solve a system of two linear equations with two variables. In this article, we will address some common questions and concerns related to solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, meaning that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know which method to use to solve a system of linear equations?

A: There are two main methods to solve a system of linear equations: substitution and elimination. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. You can choose the method that you prefer or that seems easier to you.

Q: What is the difference between the substitution and elimination methods?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. The substitution method is often easier to use when one of the equations is already solved for one variable. The elimination method is often easier to use when the coefficients of the variables are the same in both equations.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if the lines represented by the equations intersect at a single point. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.

Q: What if I get a fraction or a decimal as a solution?

A: If you get a fraction or a decimal as a solution, it means that the solution is not an integer. You can simplify the fraction or round the decimal to the nearest whole number, depending on the context of the problem.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have a built-in function to solve systems of linear equations. You can also use a graphing calculator to graph the lines represented by the equations and find the point of intersection.

Q: What if I make a mistake while solving a system of linear equations?

A: If you make a mistake while solving a system of linear equations, you can try to catch the error by checking your work. You can also try to solve the system again, using a different method or approach. If you are still having trouble, you can ask for help from a teacher or tutor.

Conclusion

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. By understanding the methods of substitution and elimination, you can solve systems of linear equations with ease. Remember to check your work and ask for help if you need it. With practice and patience, you will become proficient in solving systems of linear equations.

Frequently Asked Questions

Q: What is the difference between a system of linear equations and a system of nonlinear equations? A: A system of linear equations involves linear equations, while a system of nonlinear equations involves nonlinear equations. Nonlinear equations are equations that cannot be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: Can I solve a system of linear equations with more than two variables? A: Yes, you can solve a system of linear equations with more than two variables. However, the method of substitution and elimination may become more complex.
Q: What if I have a system of linear equations with no solution? A: If you have a system of linear equations with no solution, it means that the lines represented by the equations are parallel and do not intersect. In this case, the system has no solution.
Q: Can I use a graphing calculator to solve a system of linear equations? A: Yes, you can use a graphing calculator to solve a system of linear equations. You can graph the lines represented by the equations and find the point of intersection.

Additional Resources

Online Resources: Khan Academy, Mathway, and Wolfram Alpha are online resources that can help you solve systems of linear equations.
Textbooks: Algebra textbooks, such as "Algebra and Trigonometry" by Michael Sullivan, can provide a comprehensive introduction to solving systems of linear equations.
Tutorials: Video tutorials, such as those on YouTube, can provide step-by-step instructions on how to solve systems of linear equations.